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The motion of a compressible viscous fluid around
rotating body
Stanislav Kracmar, Sarka Necasova, Antonin Novotny
To cite this version:
Stanislav Kracmar, Sarka Necasova, Antonin Novotny. The motion of a compressible viscous fluid around rotating body. Annali dell’Universita di Ferrara, Springer Verlag, 2014, 60 (1), pp.189-209. �hal-00998361�
The motion of a compressible viscous fluid
around rotating body
S. Kraˇcmar
∗ˇ
S. Neˇcasov´a
†A. Novotn´
y
‡March 20, 2014
Department of Technical Mathematics, Czech Technical University, Karlovo n´am. 13, 121 35 Prague 2, Czech Republic
Institute of Mathematics of the Academy of Sciences of the Czech Republic, ˇ
Zitn´a 25, 115 67 Praha 1, Czech Republic
IMATH, EA 2134, Universit´e du Sud, Toulon Var, BP20132, 83957 La Garde, France
Abstract
We consider the motion of compressible viscous fluids around a rotating rigid obstacle when the velocity at infinity is non zero and parallel to the axis of rotation. We prove the existence of weak solution.
1
Viscous compressible fluid in a rotating frame
We consider a rigid body S with boundary Γ such that Ω =R3\ S is an exterior domain.
We assume that the body is rotating around the x3-axis with the angular velocity ω =
(0, 0, |ω|). The position of the body at time t > 0 is given by the formula
S(t) = {y = O(t)x; x ∈ S}, S(0) = S (1.1)
∗S. Kracmar acknowledges the support of the Czech Science Foundation (GA ˇCR) project
P201/11/1304
†S. Neˇcasov´a acknowledges the support of the Czech Science Foundation (GA ˇCR) project
P201/11/1304 in the general framework of RVO 67985840
‡The work was supported by the MODTERCOM project within the APEX programme of the region
where O(t) = cos |ω|t − sin |ω|t 0 sin |ω|t cos |ω|t 0 0 0 1 . (1.2)
We shall investigate the motion of viscous compressible fluid around the obstacle S(t). The exterior domain Ω(t) occupied by the viscous compressible fluid is time dependent:
Ω(t) = {y = O(t)x; x ∈ Ω}
Γ(t) = ∂Ω(t) = {y = O(t)x; x ∈ Γ}. (1.3) The fluid in Ω(t) is governed by the compressible Navier-Stokes equations. The unknowns, σ, v - density and velocity - solve the nonlinear system
∂t(σv) + div (v ⊗ σv) + ∇p(σ) − div S(∇v) = 0 Ω(t), t ∈ (0, ∞)
∂tσ + div (σv) = 0 in Ω(t), t ∈ (0, ∞)
v(t, y) = ω × y on Γ(t), t ∈ (0, ∞) σ(t, y) → ϱ∞, v(t, y) → a∞ as |y| → ∞,
(1.4)
completed with the initial conditions
σ(0, y) = σ0(y), v(0, y) = v0(y), y ∈ Ω.
.
Here and hereafter, S is the viscous stress tensor - a linear function of ∇xv specified
later, see (2.1), p is the pressure -a given function of the density, see (2.2), a∞ ∈ R3 is a
constant vector denoting a (prescribed) velocity of the flow at infinity and ϱ∞ > 0 is a
(prescribed) positive constant density at infinity. We omit the action of external forces for the sake of simplicity.
We shall transform the time dependent domain Ω(t) to the fixed domain Ω by em-ploying the change of variables
x = O(t)Ty, (1.5)
and introducing new functions
u(t, x) = O(t)Tv(t, y), ρ(t, x) = σ(t, y). (1.6) It is easy to verify that
O(−t) = OT(t) = O−1(t), O−1O = |ω|˙ 0 −1 0 1 0 0 0 0 0 ,
O−1O x = ω × x, ˙˙ O−1O x = −ω × x.
Moreover, the operations of inversion of the matrix O(t) and of the derivative with respect to t applied on O(t) are commuting. Using these facts, we easily find that the first term of the left-hand side in (1.4)1 reads in the new coordinates:
∂t(σv)(t, y) =
∂
∂t((O(t) ρu) (t, O
−1(t)y))
= ˙O(t)ρu((t, O−1(t)y) + O(t) ∂
t(ρu)(t, O−1(t)y) + O(t)∇x(ρu)(t, O−1(t)y
)) ˙ O−1(t)y
= O(O−1O)ρu (t, x) + O ∂˙ t(ρu) (t, x) + O ∇x(ρu) (t, x)) ˙O−1Ox
= O[∂t(ρu)−(ω×x)·∇x(ρu)+ρ(ω×u)] = O[∂t(ρu)−div (ρ(ω × x) ⊗ u)+ρ(ω×u)],
where the following standard conventions have been used
∇xu = [ aij ] i,j=1,2,3, aij = ∂ui ∂xj , (z × A)k = 3 ∑ j=1 zjAj,k, z = [z1, z2, z3], A = [ Ajk ] j,k=1,2,3, divx(z ⊗ w) = ∇x· (z ⊗ w), z ⊗ w = [ ziwk ] i,k=1,2,3. Analogously,
divy(σv ⊗ v) = O divx(ρu ⊗ u)
and
divyS(∇v)(t, y) = O divxS(∇u).
Finally, we observe that the non homogenous Dirichlet boundary conditions (1.4)3
trans-form to u = ω × x, on the fixed boundary ∂S and the conditions at infinity remain unchanged. Consequently, after the change of variables (1.5) equations (1.4) transform to
∂t(ρu) + div (ρ (u − ω × x) ⊗ u) + ρω × u + ∇p(ρ) = div S(∇u) in (0, T ) × Ω, (1.7)
∂tρ + div (ρ(u − ω × x)) = 0 in (0, T ) × Ω, (1.8)
u = ω × x on (0, T ) × ∂Ω, (1.9) u(t, x) → a∞ ∈ R3, ρ(t, x) → ρ∞ > 0 as |x| → ∞ (1.10)
This is a classical formulation of equations for unknown functions (ρ, u)(t, x), (t, x) ∈ (0, T ) × Ω that is equivalent to the formulation (1.4). We shall deal with this problem in the following sections.
We refer for more details concerning the above change of variables to Galdi [14, Chapter 1], where the calculations are performed in the case of the (incompressible) Navier-Stokes equations.
Although the problem of the moving and rotating body in a compressible fluid is of obvious practical importance, to the best of our knowledge, there is no mathematical literature about the subject. Our goal in this paper is to investigate this problem with the large initial data and in the long time run, in the context of weak solutions in the spirit introduced to the theory of incompressible Navier-Stokes equations by Leray [22] and generalized later to compressible fluids in monographs by Lions [23], Feireisl [7] and in [11], [25]. Notice, that the case without rotation has been studied in [24], [25] and the case with zero velocity at infinity has been investigated in [26].
The ”incompressible” analogue of this evolution problem is investigated in Hishida [19]. The reader can also consult Hieber and coauthors [18], and references quoted therein for the related results. The steady version of the incompressible case is investigated in Galdi and co-authors [15, 17, 16], Farwig et al. see [4, 5, 6]. Variational approach was used in Deuring et al. [3] and in [21].
It is well known from the works of P.L. Lions [23], Feireisl [7] (see also [11], [25]), that the estimates available in the compressible models do not allow to pass to the limit in the non-linear terms in the equations. Instead, one has to use the methods of compensated compactness (div-curl lemma, effective viscous flux, various commutator lemmas, oscil-lations defect measure, renormalized solutions of the transport equation) and to search strong convergence within the structure of equations. We show, that after the transfor-mation to a fixed domain one can adapt these techniques to the equations in the rotating frame. Although this choice is natural in the present context, there are many situations (e.g. domains with oscillating boundaries), where it is more convenient to work in the moving domain. We refer to [12, 13] and references quoted there, for relevant results.
Finally, notice that another situation, including Coriolis (and possibly a centrifugal force), when the fluid sticks or slips on the boundary of the body in the rotating frame and when the velocity at infinity tends to zero, enters into the framework of the exis-tence results investigated in [20]. This context is typical of the models in the physics of atmosphere; it has been therefore broadly investigated in the context of singular limits, see e.g. Bresch, Chemin, Desjardins, Feireisl, Gallagher, Grenier, G´erard-Varet, Novotny and others, [1, 2, 8, 9, 10].
The paper is organized as follows: In the first Section, we introduce the weak formu-lation and state the main result. The weak solutions are first established on large balls
with help of invading domains via a penalization method described in Section 4. In order to achieve the non zero prescribed velocity at infinity, the penalization uses an auxiliary vector field u∞, that is constructed in Section 3.The limit process on large balls with the
penalization parameter tending to infinity is performed in Section 5. Finally, the proof of the existence result is completed in Section 6, where the limit with the radius of large balls tending to infinity is established. The interesting feature in this limit process appears in the renormalized continuity equation, where the test functions have to be chosen in such a way in order to eliminate the possible contributions of the terms containing ω × x, (that explode at infinity) to the oscillations in the density sequence. This issue is handled in the last part of Section 6.
2
Setting of the problem, weak solutions
Before introducing the notion of weak solutions of the problem (1.7–1.10) and in order to determine the appropriate functional spaces, we shall specify the form of the viscous stress tensor and the pressure. Following the standard existence theory in the situation without rotation, we deal with the Newtonian fluids, where
S(∇xu) = µ T(∇xu) + η divxu I, T(∇xu) = ∇xu + ∇T
xu −
2
3divxu I, (2.1) with µ > 0, η ≥ 0. Moreover, we assume that the pressure is a given function of the density that satisfies hypotheses
p ∈ C[0, ∞) ∩ C1(0, ∞), p(0) = 0, p′(ϱ) > 0 for all ϱ > 0, lim
ϱ→∞
p′(ϱ)
ϱγ−1 = a > 0 (2.2)
for a certain γ > 3/2. Notice that this assumption covers at least one physically reasonable case, namely isentropic flows of a monoatomic gas.
In view of these assumptions, the weak formulation of problem (1.7–1.10) reads:
Definition 1
We shall call a couple (ρ, u) a renormalized weak solution of problem (1.7–2.2) if: (i) Functions (ϱ, u) are such that
ρ ≥ 0, ρ − ϱ∞ ∈ L∞(0, T ; Lγ+ L2(Ω)),
ϱ(u − a∞)2 ∈ L∞(0, T ; L1(Ω)),
u − a∞∈ L2(0, T ; W1,2(Ω)), u(t)|∂Ω = ω × x for a.a. t ∈ (0, T ).
(ii) Density ϱ ∈ Cweak([0, T ]; Lγ(K)) for any compact K ⊂ Ω and the equation of
conti-nuity (1.8) is satisfied in the weak sense, ∫ Ω ϱφdx τ 0 = ∫ τ 0 ∫ Ω ϱ∂tφ + ϱ(u − ω × x) · ∇xφ dx dt, (2.4)
for all τ ∈ [0, T ] and any test function φ ∈ C∞
c ([0, T ] × Ω), as well as in the
renormalized weak sense ∫ Ωb(ϱ)φdx τ 0 = ∫τ 0 ∫ Ωb(ϱ) ( ∂tφ + (u − ω × x) · ∇xφ ) dx dt +∫τ 0 ∫ Ω ( b(ϱ) − ϱb′(ϱ))divu φ dx dt (2.5)
for any τ ∈ [0, T ], for any φ ∈ C1
c([0, T ] × Ω) and for all
b ∈ Cc[0, ∞) ∩ Cc1(0, ∞) (2.6)
Here and in what follows ∫
Ωgφdx τ 0 = ∫ Ωg(τ, x)φ(τ, x)dx − ∫ Ωg0(x)φ(0, x)dx.
(iii) The linear momentum ϱu ∈ Cweak([0, T ], L2γ/(γ+1)(K)) for any compact K ⊂ Ω and
the momentum equation (1.7) is satisfied in the weak sense ∫ Ωϱu · φdx τ 0 + ∫ τ 0 ∫ Ω S(∇xu) : ∇xφ dx dt = ∫ τ 0 ∫ Ω (
ϱu · ∂tφ + ϱ(u − ω × x) ⊗ u : ∇xφ − ϱ(ω × u) · φ + p(ϱ) divxφ
)
dxdt (2.7)
for all τ ∈ [0, T ] and for any test function φ ∈ C∞
c ([0, T ] × Ω).
Notice that the class (2.6) of the renormalizing functions b can be enlarged to b ∈ C[0, ∞) ∩ C1(0, ∞), z b′(z) ∈ L∞(0, 1), b(z)/z5γ/6, (zb′− b)(z)/zγ/2∈ L∞(1, ∞)
(2.8) by using the Lebesgue dominated convergence theorem.
Here and hereafter, we denote
H(ρ) = ρ ∫ ρ 1 p(s) s2 ds, (2.9) E(ϱ|ϱ∞) = H(ϱ) − H′(ϱ∞)(ϱ − ϱ∞) − H(ϱ∞). (2.10)
We are now in a position to state the main existence result.
Theorem (Existence of weak solutions) Assume that ρ∞ > 0 and Ω = R3 \ S,
where S is a bounded Lipschitz domain. Suppose that the vector ω is parallel to the vector a∞, and that the viscous stress tensor S and the pressure p satisfy (2.1), (2.2). Assume
that
ϱ0 ≥ 0, E(ρ0|ϱ∞) ∈ L1(Ω) ϱ0(u0− a∞)2 ∈ L1(Ω). (2.11)
Then the problem (1.7–1.11) admits at least one renormalized weak solution.
3
An auxiliary result
Let us denote the open ball with radius r centered at the origin by Br. Suppose without
loss of generality that S ⊂ B1/2 and B1 ⊂ BR. To begin, we claim that there exists a
smooth vector field u∞ ∈ Cc∞(R3) such that:
u∞ = ω × x in S supp u∞⊂ B2R a∞ in B(3/2)R\ B1 (3.1) div u∞ = 0 in S ∪ (R3\ B1) (3.2)
Let us consider non-increasing functions Φ1, Φ2, ΨR∈ Cc∞([0, +∞)) as follows: 0 ≤ Φ1 ≤
1, Φ1 = 1 in [ 0, 5/9) , Φ1 = 0 in [2/3, +∞) , 0 ≤ Φ2 ≤ 1, Φ2 = 1 in [ 0, 7/9) , Φ2 = 0 in
[ 8/9, +∞) , 0 ≤ ΨR ≤ 1, ΨR= 1 in [ 0, (3/2)R)) , ΨR= 0 in [2R − 1/4, +∞) . Let us set
˜
a∞(x) = a∞(1 − Φ2(|x|))ΨR(|x|).
Using the classical Stokes formula it is easy to check that ∫
B2R−1/4\B(3/2)R
div ˜a∞(x) dx = 0.
So, from Lemma 3.1 (see further) we get that there exists a vector field b∞ ∈
W01,p(B2R−1/4\ B(3/2)R), p ∈ (1, +∞) and
div b∞(x) = div ˜a∞(x) for x ∈ B2R−1/4\ B(3/2)R.
It is easy to see that divergence of the expression (ω × x) Φ1(|x|) ∈ Cc∞(R3) is zero. Now,
let extend b∞(·) by zero in R3 and set
where 0 < ε < 1/8 is fixed and mε(·) is the standard mollifier. In the previous reasoning,
we have used the auxiliary result concerning the so called Bogovskiˇı operator (see e.g. Galdi [15]):
Lemma 3.1. Let G ⊂ RN be a bounded domain with Lipschitz boundary. Then there
exists a linear operator BG= (B(1)G , B(2)G , . . . B(N )G ) with the following properties:
1. BG: { F ∈ Lq(G); ∫ GF dx = 0 } −→ W01,q(G;RN )
is a bounded linear operator. More precisely, for any q ∈ (1, ∞), ∥∇BG(F)∥Lq(G;RN ×N)≤ c(G, q)∥F∥Lq(G);
2. the function v = BG(F) solves the problem
div v = F in G, v|∂G = 0;
3. if, moreover, F can be written in the form F = div Q for a certain Q ∈ Lp(G,RN)
satisfying Q · n|∂G = 0 where n stands for the outward unit normal vector to ∂G,
then it holds
∥BG(F)∥Lp(G;RN) ≤ c(G, p)∥Q∥Lp(G).
4
Penalized problem
In what follows, we denote
V = B2R
and extend ϱ0 by 0 and u0 by ω × x in S. We shall solve on (0, T ) × V the following
penalized problem
∂t(ρu) + div ((ρ(u − ω × x) ⊗ u)) + ρω × u − div S(∇u)
+∇p(ρ) + k · 1{(V \BR)∪S}(u − u∞) = 0 (4.1) ∂tρ + div (ρ(u − ω × x)) = 0. (4.2) u|∂V = 0, (4.3) ϱ(0) = ϱ0, ϱu(0) = ϱ0u0. (4.4)
Definition 4.1. A couple (ϱ, u) is a renormalized finite energy weak solution of the pe-nalized problem (4.1–4.4) if (i) ρ ≥ 0, ρ − ϱ∞ ∈ L∞(0, T ; [Lγ+ L2](V )), ϱ(u − u∞)2 ∈ L∞(0, T ; L1(V )), u − u∞∈ L2(0, T ; W01,2(V ;R3)). (4.5)
(ii) Equation of continuity (1.8) is satisfied in the weak sense, see (2.4) with Ω replaced by V for any test function φ ∈ C∞
c ([0, T ] × V ), as well as in the renormalized weak
sense, see (2.5) with Ω replaced by V for any b as in (2.5) and any φ ∈ C1
c([0, T )×V ).
(iii) Momentum equation is satisfied in the weak sense , ∫ τ
0
∫
Ω
(
ϱu·∂tφ+ϱ(u−ω×x)⊗u : ∇xφ−ϱ(ω×u)·φ−k·1{(V \BR)∪S}(u−u∞)·φ (4.6)
+p(ϱ)divxφ ) dx dt = ∫ Ω ϱuφ dx τ 0 + ∫ τ 0 ∫ Ω S(∇xu) : ∇xφ dx dt
for all τ ∈ [0, T ] and for any test function φ ∈ C∞
c ([0, T ] × V ).
(iv) The following energy inequality is satisfied
∫ V ( 1 2ϱ|u − u∞| 2+ E(ϱ|ϱ ∞) ) dx τ 0 +∫τ 0 ∫ V ( S(∇x(u − u∞)) : ∇x(u − u∞) + k1(V \B R)∪S|u − u∞| 2)dx dt ≤ −∫τ 0 ∫ V p(ρ)div u∞+ ∫τ 0 ∫ V ρ(u − ω × x) · ∇xu∞· (u∞− u) dx dt −∫τ 0 ∫ V ϱ(ω × u∞) · (u − u∞) dx dt − ∫τ 0 ∫ V S(∇xu∞) : ∇x(u − u∞) dx dt (4.7) for a. a. τ ∈ (0, T ).
Under assumptions (2.11) with u∞ defined in formulas (3.1–3.2), problem (4.1–4.2)
admits at least one weak solution (ϱk, uk). This result can be proved similarly as in
5
Limit k → ∞
We deduce from estimate (4.7) the following estimates, with c = c(R) a generic constant that may depend on R but is independent of k:
ess sup t∈(0,T ) ∫ V ρk|uk− u∞|2dx ≤ c(R) (5.1) ess sup t∈(0,T ) ∫ V E(ρk|ρ∞)dx ≤ c(R) (5.2) ∥uk∥L2(0,T ;W1,2(V ;R3))≤ c(R) (5.3) ∥uk− u∞∥L2((0,T )×((V \BR)∪S);R3)≤ c(R) √ k (5.4)
When deducing (5.3) we have used the Korn-Poincar´e type inequality in the form ∥S (∇(uk− u∞))∥L2(V ;R3×3) ≥ c∥uk− u∞∥
W01,2(V ;R3×3), (5.5)
see Theorem 10.16 in [11]. Finally, introducing for any h ∈ L1((0, T ) × V )
[h]ess= 1{|ρ−ρ∞|≤ρ∞/2}· h, [h]res= h − hess, (5.6)
and taking into account
E(ρ|ρ∞) ≥ c(|ρ − ρ∞|2· 1{ρ∞/2≤ρ≤2ρ∞}+ (1 + ρ
γ
) · 1{ρ<ρ∞/2 or ρ>2ρ∞}
) (see e.g. [11]) we get from (5.2):
ess sup t∈(0,T )∥[ρk− ρ∞]ess∥L2(V ) ≤ c(R),
ess sup t∈(0,T )∥[ρk]res∥Lγ(V ) ≤ c(R). (5.7)
On the basis of these estimates we get the existence of a subsequence (ϱk, uk) and a
couple (ϱ, u) such that
ϱk− ϱ∞ ⇀ ϱ − ϱ∞ in L∞(0, T ; [L2+ Lγ](V ))
uk ⇀ u in L2(0, T ; W1,2(V, R3)),
where ϱ ≥ 0. It can be proved via quite recent but nowadays classical methods (including effective viscous flux, oscillations defect measures, renormalized solutions of continuity equation) [23], [7], [25], [11] that the couple (ϱ, u) has the following properties 1:
(i) ϱ ∈ Cweak(0, T ; Lγ(V )) and the continuity equation is satisfied in the weak sense ∫ V ϱφdx τ 0 = ∫ τ 0 ∫ V ϱ∂tφ + ϱ(u − ω × x) · ∇xφdx dt (5.8)
for all τ ∈ [0, T ] and any φ ∈ C∞
c ([0, T ] × V ). Similarly, the renormalized continuity
equation is satisfied in the form: ∫ V b(ϱ)φdx τ 0 + ∫ τ 0 ∫ V ( ϱb′(ϱ) − b(ϱ))divuφdxdt = ∫ τ 0 ∫ V b(ϱ)(∂tφ + (u − ω × x) · ∇xφ ) dxdt (5.9)
for all τ ∈ [0, T ], for any b belonging to the class (2.6) and for any φ ∈ Cc1([0, T ] × V ).
(ii) Since u − ω × x = 0 a.e. in (0, T ) × S, equation (5.8) yields, in particular, ∫ T 0 η′(t) ∫ S ϱφ(x)dxdt = 0
for all η ∈ C1(0, T ) and for all φ ∈ C1
c(S). This means that
for a.a. (t, x) ∈ (0, T ) × S, ϱ(t, x) = ϱ0(x) = 0.
With this information at hand, formula (5.8) yields, in particular, ∫ ΩR ϱφdx τ 0 = ∫ τ 0 ∫ ΩR ϱ(∂tφ + (u − ω × x) · ∇xφ ) dx dt (5.10)
for all τ ∈ [0, T ] and for any test function φ ∈ C∞
c ([0, T ] × (BR\ S)). Here and in
what follows, we denote ΩR= Ω ∩ BR= BR\ S. Similarly, formula (5.9) yields
∫ ΩR b(ϱ)φdx τ 0 = ∫ τ 0 ∫ ΩR ( b(ϱ) − ϱb′(ϱ))divuφdxdt ∫ τ 0 ∫ ΩR b(ϱ)(∂tφ + (u − ω × x) · ∇xφ ) dxdt (5.11)
for all τ ∈ [0, T ], for any b belonging to the class (2.6) and φ ∈ C1
(iii) There holds ϱu ∈ Cweak(0, T ; L2γ/γ+1(V ; R3)) and ∫ T 0 ∫ ΩR (
ϱu · ∂tφ + ϱ(u − ω × x) ⊗ u : ∇xφ − ϱ(ω × u) · φ + p(ϱ)divxφ
) dx dt (5.12) = ∫ ΩR ϱuφdx τ 0 + ∫ T 0 ∫ ΩR S(∇xu) : ∇xφdx dt
for any τ ∈ [0, T ] and all test functions φ ∈ C∞
c ([0, T ] × ΩR). Moreover,
u = u∞a.e. in (0, T ) × [(V \ BR) ∪ S]. (5.13)
(iv) The following energy inequality is satisfied
∫ ΩR ( 1 2ϱ|u − u∞| 2+ E(ϱ|ϱ ∞) ) dx τ 0 +∫τ 0 ∫ ΩR( S(∇x(u − u∞)) : ∇x(u − u∞)dxdt ≤ −∫τ 0 ∫ B1\Sp(ρ)divu∞− ∫τ 0 ∫ B1\Sρ(u − ω × x) · ∇xu∞· (u − u∞)dxdt −∫τ 0 ∫ B1\Sϱ(ω × u∞) · (u − u∞) dx dt − ∫τ 0 ∫ B1\S S(∇xu∞) : ∇x(u − u∞)dxdt (5.14) for a. a. τ ∈ (0, T ).
When deriving (5.14) we have used the facts that: 1) ω and u∞ are parallel if x ∈
B3
2R\ B1 and 2) properties (3.1), (3.2) and (5.13).
6
Limit R → ∞
6.1
Weak limits
We set U∞ = { u∞ in B1, a∞ in R3\ B1 } . (6.1)Here and hereafter we extend ρ by ρ∞ outside BR (and denote the new function by ϱ),
and u by U∞ outside BR and denote the new function again by u. Using (5.1-5.7) we
∫
R3
[1]resdx ≤ c (6.2)
ess sup t∈(0,T )∥[ρR− ρ∞]ess∥L2(R3) ≤ c (6.3)
ess sup t∈(0,T )∥[ρR]res∥Lγ(R3) ≤ c (6.4)
ess sup t∈(0,T )∥ϱR|uR− U∞|2∥L1(R3) = 0. (6.5)
∥uR− U∞∥L2((0,T )×W1,2(R3;R3))≤ c (6.6)
Hereafter, we describe in a concise form the nowadays classical argument including effective viscous flux identity, oscillations defect measures and renormalized continuity equation leading to the proof of the main theorem. It is to be noticed that the main issue is to show the strong convergence of the density sequence.
We use in momentum equation (5.12) (written with (ϱR, ϑR, uR) on ΩR) with the test
function φ = BΩn(ϱ
ν
R) ∈ L∞(0, T ; W 1,γ/ν
0 (Ωn; R3)) ∩ L2(0, T ; Lp(Ωn)), νγ = 12 = 1p, where
ν > 0 is sufficiently small and n ∈ N is fixed. After some technical calculations similar to [11, Section 2.2.5, Appendix], where one uses the Bogovskii lemma 3.1, one derives that there exists ν > 0 such that for all R ≥ n,
∫ T
0
∫
Ωn
ϱγ+νR dxdt ≤ c(n). (6.7)
Proceeding as in [25, Section 7.12.6] we deduce from estimates (6.2–6.7), by means of a diagonalisation procedure, existence of a couple (ϱ, u) such that
ϱR− ϱ∞⇀∗ ϱ − ϱ∞ in L∞(0, T ; Lγ+ L2(Ω)),
uR− U∞ ⇀ u − U∞in L2(0, T ; W1,2(Ω; R3)),
p(ϱR) ⇀ p(ϱ) in Lq((0, T ) × Ωn) with some q > 1, ∀n ∈ N,
(6.8)
as R → ∞. Here and hereafter g(ϱ, u) denotes L1 weak limit of the sequence g(ϱ
R, uR).
Moreover, using continuity equation (5.10), renormalized continuity equation (5.9) and momentum equation (5.12) together with estimate (6.3–6.5) (all written with (ϱR, uR) on
ΩR)), we get, after employing the Arzela-Ascoli compactness argument and a density
argument,
ϱR → ϱ in Cweak([0, T ]; Lγ(Ωn)) and in L2(0, T ; W−1,2(Ωn)), ∀n ∈ N,
h(ϱR) → h(ϱ) in Cweak([0, T ]; Lq(Ωn)), q ∈ [1, ∞) and in L2(0, T ; W−1,2(Ωn)), ∀n ∈ N,
where h belongs to (2.6),
ϱRuR→ ϱu in Cweak([0, T ]; L2γ/(γ+1)(Ωn; R3)) and in L2(0, T ; W−1,2(Ωn, R3)), ∀n ∈ N,
ϱRuR⊗ uR⇀ ϱu ⊗ u in L2(0, T ; L6γ/(4γ+3)(Ωn, R3×3)), ∀n ∈ N.
(6.9) In all three cases, the strong L2W−1,2 convergence follows from the previous C
weakLq
convergence and the compact embedding Lq(G) ֒→֒→ W−1,2(G), q > 6/5, where G is a
bounded domain.
At this stage, we may pass to the limit R → ∞ in equations (5.12), (5.10) and (5.9) that are written with (ϱR, ϑR, uR) on ΩR. We get
∫ Ω ϱφ dx τ 0 = ∫ τ 0 ∫ Ω ( ϱ∂tφ + ϱ(u − ω × x) · ∇xφ ) dx dt (6.10)
for all τ ∈ [0, T ] and for any φ ∈ C1
c([0, T ] × Ω); ∫ Ωϱu · φ dx τ 0 (6.11) = ∫ τ 0 ∫ Ω (
ϱu · ∂tφ + ϱ(u − ω × x) ⊗ u : ∇xφ + p(ϱ)divxφ −S(∇xu) : ∇xφ
) dx dt
for all τ ∈ [0, T ] and for any φ ∈ C1
c([0, T ] × Ω; R3); ∫ τ 0 ∫ Ω ϱLk(ϱ) ( ∂tφ + (u − ω × x) · ∇xφ ) dxdt (6.12) = ∫ τ 0 ∫ Ω Tk(ϱ)divuφdxdt + ∫ Ω ϱLk(ϱ)φdx τ 0 and ∫ τ 0 ∫ Ω Tk(ϱ) ( ∂tφ + (u − ω × x) · ∇xφ ) dxdt (6.13)
= ∫ τ 0 ∫ Ω (ϱT′ k(ϱ) − Tk(ϱ))divuφdxdt + ∫ Ω Tk(ϱ)φdx τ 0, where τ ∈ [0, T ] and φ ∈ C1
c([0, T ) × Ω). In the above formulas, we have taken
Tk(z) = kT (z/k), Lk(z) =
∫ z
1
Tk(w)
w2 dw, ∀z ≥ 0, T (z) = min{z, k}.
6.2
Strong convergence of densities
6.2.1 Effective viscous fluxUsing the same arguments as in [11, Chapter 3, Section 3.7] we can show that
Tk(ϱ)divxu − Tk(ϱ)divxu = 1 4 3µ + η ( p(ϱ)Tk(ϱ) − p(ϱ) Tk(ϱ) ) . (6.14)
To get this identity, one needs to subtract the limit R → ∞ of the momentum equation (5.13) (written on ΩR with (ϱR, uR)) tested with φ = ζ∇∆−1[Tk(ϱR)ζ],
ζ ∈ C∞((0, T ) × Ω
n; R3) from the limiting momentum equation (6.11) with the
test function φ = ζ∇∆−1[T
k(ϱ)ζ], where n is fixed. Here and hereafter we
de-note (∇∆−1) j(v) = −F−1 [iξ j |ξ|2F(v)(ξ) ]
(that is a continuous operator from Lp(R3) to
{v ∈ L3p/(3−p)(R3; R3), ∇v ∈ Lp(R3; R3×3)}, 1 < p < 3 and R the Riesz operator,
(R[v])ij = (∇ ⊗ ∇∆−1)ijv = F−1
[
ξiξj
|ξ|2F(v)(ξ)
]
with F the Fourier transform. It is im-portant to recall that R is a continuous operator from Lp(R3) → Lp(R3, R3×3) for any
1 < p < ∞.
SinceS is linear, this procedure yields
lim R→∞ ∫ T 0 ∫ Ω ζ(t, x)(p(ϱR)Tk(ϱR) −S(∇xuR) : R[ζTk(ϱR)] ) dxdt = ∫ T 0 ∫ Ω ζ(t, x)(p(ϱ) Tk(ϱ) −S(∇xu) : R[ζTk(ϱ)] ) dxdt lim R→∞ ∫ T 0 ∫ Ω (uR− ω × x) · ( ζTk(ϱR)R[ζϱRuR] − ζϱRuRR[ζTk(ϱR)] ) dxdt − ∫ T 0 ∫ Ω (uR− ω × x) · ( ζTk(ϱ)R[ζϱu] − ζϱuR[ζTk(ϱ)] ) dxdt, (6.15)
where (R[z])j =∑3k=1Rjk[zk]. In fact, other (lower order) terms that should eventually
appear at the right hand side are zero by virtue of a standard compactness argument. At this stage, we need a div-curl type lemma in the form [11, Theorem 10.27], stating
Lemma 6.1. Let UR⇀ U in Lp(R3; R3), vR ⇀ v in Lq(R3) as R → ∞, where
1 p + 1 q = 1 r < 1. Then
vRR[UR] − R[vR]UR ⇀ vR[U] − R[v]U
in Lr(R3; R3).
Combining this lemma with the convergence established in the first two lines of (6.9), we get ( ζTk(ϱR)R[ζϱRuR] − ζϱRR[ζTk(ϱR)]uR ) (t) →(ζTk(ϱ)R[ζϱu] − ζϱR[ζTk(ϱ)]u ) (t)
weakly in Lr(R3; R3) with some r > 6/5 for all t ∈ [0, T ] as R →∞; whence by the
compact imbedding Lr(Ω
n) ֒→֒→ W−1,2(Ωn) and the Lebesgue dominated convergence
theorem used over (0, T ) we conclude that the r.h.s. of (6.15) is equal to 0. This yields the effective viscous flux identity (6.14).
6.2.2 Oscillations defect measure and renormalized continuity equation Due to (2.2),
p(ϱ) = dϱγ+ pm(ϱ), for some d > 0, where ∂ϱpm(ϱ) ≥ 0. (6.16)
We have d lim sup R→0 ∫ T 0 ∫ Ωζ|T k(ϱR) − Tk(ϱ)|γ+1dxdt ≤ d lim sup R→∞ ∫ T 0 ∫ Ω ζ((Tk(ϱR) − Tk(ϱ) ) (ϱγR− ϱγ)dxdt ≤ (6.17) ∫ T 0 ∫ Ω ζ(ϱγT k(ϱ)− ϱγ Tk(ϱ) ) dxdt ≤ ∫ T 0 ∫ Ω ζ(p(ϱ) Tk(ϱ)− p(ϱ) Tk(ϱ) ) dxdt, where ζ ∈ C∞
c ((0, T ) × Ω), ζ ≥ 0. The first inequality is an algebraic one, to derive the
second one, we have used convexity of ϱ → ϱγ and concavity of ϱ → T
derive the third one, we have employed the standard relation between the weak limits of monotone functions that reads
pm(ϱ)Tk(ϱ) − pm(ϱ) Tk(ϱ) ≥ 0, (6.18)
cf. e.g. [11, Theorem 10.19].
The right hand side of inequality (6.17) can be calculated from the effective viscous flux identity (6.14); using (6.6) we find that
∫ T 0 ∫ Ωn |Tk(ϱR) − Tk(ϱ)|γ+1dxdt ≤ c(n)∥Tk(ϱR) − Tk(ϱ)∥L2((0,T )×Ωn) ≤ ∥Tk(ϱR) − Tk(ϱ)∥(γ−1)/(2γ)L1((0,T )×Ωn)∥Tk(ϱR) − Tk(ϱ)∥(γ+1)/(2γ)Lγ+1((0,T )×Ωn); whence oscγ+1[ϱR → ϱ]((0, T )×Ωn) ≡ sup k>0 lim sup R→∞ ∫ T 0 ∫ Ωn |Tk(ϱR)−Tk(ϱ)|γ+1dxdt ≤ c(n), (6.19)
where the expression at the left hand side is called oscillations defect measure, see Feireisl [7].
On the other hand, relation (6.19) implies that the limit quantities ϱ, u satisfy the renormalized equation of continuity (2.5), see Feireisl [7],
Lemma 6.2 (Renormalized continuity equation). Let G ⊂ R3 be open set and let
ϱR ⇀ ϱ in L1((0, T ) × G),
uR⇀ u in Lr((0, T ) × G; R3),
∇uR ⇀ ∇u in Lr((0, T ) × G; R3×3), r > 1.
Let
oscq[ϱR→ ϱ](G) < ∞ (6.20)
for 1q < 1 − 1r, where (ϱR, uR) solves the renormalized continuity equation (2.5) (with Ω
replaced by G). Then the limit functions ϱ, u solve (2.5) with (Ω replaced by G) as well.
6.3
Strong convergence of density sequence
We deduce using Lemma 6.2 that∫ τ 0 ∫ Ω ϱLk(ϱ) ( ∂tφ + (u − ω × x) · ∇xφ ) dxdt (6.21)
= ∫ τ 0 ∫ Ω Tk(ϱ)divuφdxdt+ ∫ Ω ϱLk(ϱ)φdx τ 0,
where Lk(ϱ) is defined in (6.13), τ ∈ [0, T ] and φ ∈ Cc∞([0, T ] ×Ω). Reasoning as in (6.18)
we verify that
p(ϱ)Tk(ϱ) − p(ϱ) Tk(ϱ) ≥ 0; (6.22)
whence subtracting (6.12), (6.21) and noticing that (ϱ log ϱ− ϱ log ϱ)(0) = 0, we deduce after letting k → ∞, ∫ Ω ( ϱ log ϱ − ϱ log ϱ)(τ )ΦR(x) (6.23) ≤ ∫ τ 0 ∫ Ω ( ϱ log ϱ − ϱ∞log ϱ∞ ) (u − ω × x) · ∇xΦR(x)dxdt − ∫ τ 0 ∫ Ω ( ϱ log ϱ − ϱ∞log ϱ∞ ) (u − ω × x) · ∇xΦR(x)dxdt.
Here, we have used (6.22), and we have chosen ΦR(x) = η(x3/Rα)ϕ(|x′|/R), R > 1,
α > 0, x′ = (x 1, x2), η ∈ Cc∞(R), ϕ ∈ Cc∞(R), 0 ≤ η, ϕ ≤ 1, ϕ(|x′|) = ϕ(|x′|) = 1 if |x′| ≤ 1 ϕ(|x′|) = 0 if |x′| ≥ 2 , η(s) = η(s) = 1 if |s| ≤ 1 η(s) = 0 if |s| ≥ 2 .
It is easy to check that
∥∇xΦR∥L∞(R3;R3) → 0 as R → ∞, ∥∂x3ΦR∥q Lq(R3) ≤ cR2+α−αq, 1 ≤ q < ∞, and that (ω × x) · ∇xΦR(x) = 0. Consequently, ∫ Ω ( ϱ log ϱ − ϱ log ϱ)(τ )ΦR(x) (6.24) ≤ ∫ τ 0 ∫ Ω ϱ log ϱ − ϱ∞log ϱ∞ U∞ ∂x3ΦR(x) dxdt − ∫ τ 0 ∫ Ω ϱ log ϱ − ϱ∞log ϱ∞ |U∞| ∂x3ΦR(x) dxdt, + ∫ τ 0 ∫ Ω ( ϱ log ϱ − ϱ∞log ϱ∞ ) (u − U∞) · ∇xΦR(x)dxdt
+ ∫ τ 0 ∫ Ω ( ϱ log ϱ − ϱ∞log ϱ∞ ) (u − U∞) · ∇xΦR(x)dxdt.
Due to uniform bounds (6.2-6.6), the right hand side of this inequality tends to 0 as R → ∞, provided we chose α > 2. Consequently,
ϱ log ϱ − ϱ log ϱ = 0; whence
ϱR→ ϱ a.e. in (0, T ) × Ω.
Remark 6.1
It is to be noticed that the couple (ϱ, u) satisfies the energy inequality in the form
∫ Ω ( 1 2ϱ|u − U∞| 2+ E(ϱ|ϱ ∞) ) dx τ 0 +∫τ 0 ∫ Ω(S(∇x(u − U∞)) : ∇x(u − U∞)dxdt ≤ −∫τ 0 ∫ B1\Sp(ρ)divU∞− ∫τ 0 ∫ B1\Sρ(u − ω × x) · ∇xU∞· (u − U∞)dxdt −∫τ 0 ∫ B1\Sϱ(ω × U∞) · (u − U∞) dx dt − ∫τ 0 ∫ B1\S S(∇xU∞) : ∇x(u − U∞)dxdt (6.25) for a. a. τ ∈ (0, T ), where U∞ is defined in (6.1).
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